Part (REF ) follows from [1]} for \(n=0\)  and [2]} for \(n>0\) .
Part (REF ) follows from [3]}. Part (REF ) follows from [4]} together with Remark REF .
The proof of part (REF ) follows by the following cycle of implications:
(REF ) \(\Rightarrow \)  (REF ) follows from [5]} and Remark REF .
(REF ) \(\Rightarrow \)  (REF ) follows from the fact that \(\overline{\mathcal {M}}_{g,n}\)  is proper.
(REF ) \(\Rightarrow \)  (REF ) follows from [3]} using that a separated algebraic stack has finite stabilizers at every geometric point.
(REF ) \(\Rightarrow \)  (REF ) follows from [7]} and Remark REF .
Part (REF ) follows since the obstruction to deform a line bundle on a quasi-stable curve \(C\)  lies on \(H^2(C,\mathcal {O}_C)\) , which is zero because \(C\)  has dimension one.
Part (REF ) follows from part (REF ) and the well-known fact that \(\overline{\mathcal {M}}_{g,n}^{qs}\)  is smooth and that \(\partial \overline{\mathcal {M}}_{g,n}^{qs}:=\overline{\mathcal {M}}_{g,n}^{qs}\setminus \mathcal {M}_{g,n}\)  is a normal crossing divisor. Alternatively, one could use the explicit description of the completed local rings of \(\overline{\mathcal {J}}_{g,n}\)  given in [3]} or [9]}

